Quadratically Converging Algorithms for it

We present a quadratically converging algorithm for m based on a formula of Legendre's for complete elliptic integrals of modulus sin(w/12) and the arithmetic-geometric mean iteration of Gauss and Legendre. Precise asymptotics are provided which show this algorithm to be (marginally) the most efficient developed to date. As such it provides a natural computational check for the recent large-scale calculations of m. 1. The Algorithms. The arithmetic-geometric mean of Gauss and Legendre is defined, for k e (0,1], by (1) an+x = "2 ", bn + x = ]ja~X, cn+x = -(a„ b„) with a0:= 1, b0:= \1 k2 := k', c0:= k. The common limit of {a„} and {/>„} we call AGM(ri'). The remarkable utility of the above iteration stems from two observations. Firstly, 0 <K< bn + i <«„+! <a„ and cn+x = c2/4an+x, which show that both sequences converge quadratically. Secondly, their common limit can be expressed in terms of complete elliptic integrals of the first kind K:= K(k), that is, (2)-= f dt 2AGM(*') ~ h Jl-k2sm2t K. Complete elliptic integrals of the second kind E:=E(k):= P^Vl k2un21 dt can also be calculated from the arithmetic-geometric mean iteration. Precisely, (3) (K-E)/K=l/2(c2 + 2c2+ ••• +2"c„2 + •••). This powerful tool for computing elliptic integrals can be used to derive algorithms for 7T as follows. Let E' := E(k') and K':= K(k'). (These are the complete elliptic integrals in the conjugate modulus k' = vl k2.) Then, Legendre's formula relating these quantities is (4) EK' + E'K KK' = \ 77. Received May 24, 1983; revised January 23, 1984 and September 25, 1984. 1980 Mathematics Subject Classification. Primary 65D20, 65C25, 32A25, 10F05.